"In a metamathematical sense our theory provides general concepts applicable to all branches of abstract mathematics, and so contributes to the current trend towards uniform treatment of different mathematical disciplines. In particular, it provides opportunities for the comparison of constructions and of the isomorphisms occurring in different branches of mathematics; in this way it may occasionally suggest new results by analogy." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)
"The invariant character of a mathematical discipline can be formulated in these terms. Thus, in group theory all the basic constructions can be regarded as the definitions of co- or contravariant functors, so we may formulate the dictum: The subject of group theory is essentially the study of those constructions of groups which behave in a covariant or contravariant manner under induced homomorphisms." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)
"The subject of group theory is essentially the study of those constructions of groups which behave in a covariant or contravariant manner under induced homomorphisms. More precisely, group theory studies functors defined on well specified categories of groups, with values in another such category." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)
"The theory [of categories] also emphasizes that, whenever new abstract objects are constructed in a specified way out of given ones, it is advisable to regard the construction of the corresponding induced mappings on these new objects as an integral part of their definition. The pursuit of this program entails a simultaneous consideration of objects and their mappings (in our terminology, this means the consideration not of individual objects but of categories). This emphasis on the specification of the type of mappings employed gives more insight onto the degree of invariance of the various concepts involved." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)
"Category theory is an embodiment of Klein’s dictum that it is the maps that count in mathematics. If the dictum is true, then it is the functors between categories that are important, not the categories. And such is the case. Indeed, the notion of category is best excused as that which is necessary in order to have the notion of functor. But the progression does not stop here. There are maps between functors, and they are called natural transformations." (Peter Freyd, "The theories of functors and models", 1965)
"Categorical algebra has developed in recent years as an effective method of organizing parts of mathematics. Typically, this sort of organization uses notions such as that of the category G of all groups. [...] This raises the problem of finding some axiomatization of set theory - or of some foundational discipline like set theory - which will be adequate and appropriate to realizing this intent. This problem may turn out to have revolutionary implications vis-`a-vis the accepted views of the role of set theory." (Saunders Mac Lane, "Categorical algebra and set-theoretic foundations", 1967)
"The theory of categories has arisen in the last twenty-five years and now constitutes an autonomous branch of mathematics. It owes its origin and early inspiration to developments in algebraic topology. When the basic concepts of category, functor, natural transformation and natural equivalence were first formulated by Eilenberg and Mac Lane they served immediately to provide the appropriate framework for describing the way in which algebraic tools were used, and could be used, in the study of topology. It was surely evident from the outset, to the inventors of these fundamental notions and to others, that their domain of application certainly extended far beyond that of algebraic topology. […] However, it was not clear in the early stages that there was a “pure” theory latent within the domain of categories and functors which was capable of assuming substantial proportions within the body of mathematics. […] Nevertheless, it is only in the last ten years, or less, that the source of inspiration for advances in category theory has come to any considerable extent from within the theory itself." (Ion Bucur & Aristide Deleanu, "Introduction to the Theory of Categories and Functors", 1968)
"Yet, unless I am not sufficiently aware of current trends, the set-theoretical difficulties in handling categories have not inspired many set theorists and it has had little impact in logic as a whole. Thus, although we thoroughly accepted highly impredicative set theory because we understand its internal cogency, we, as logicians, are less likely to accept category theory whose roots lie in algebraic topology and algebraic geometry. It could be retorted that the existing axioms of infinity are ample to cover formalizations of category theory, yet an obstinate categorist could say that categories themselves should be accepted as primitive objects." (Paul J Cohen, "Comments on the foundations of set theory", 1971)
"A theory is a category with certain operations (defined up to isomorphism) [...] The notion of theory is thus 'intrinsic', i.e., independent of a particular presentation via formal languages and axiomatic systems. In this sense, categorical logic may be viewed as 'synthetic' or 'intrinsic' logic by opposition to the usual 'analytic', 'formal' logic." (André Joyal & Gonyalo E Reyes, "Forcing and generic models in categorical logic", 1977)
"But the question automatically arises as to exactly why, in introducing ordinary mathematical notions into the theory, one must make a detour through the somewhat opaque notion of discrete category. It is difficult to see how this can be explained except by appeal to the notion of 'unstructured' category, i.e. set." (John L Bell, "Category theory and the foundations of mathematics", The British Journal for the Philosophy of Science 32(4), 1981)
"But this seems to me highly dubious, for it is surely the case that the unstructured notion of class is epistemically prior to any more highly structured notion such as category: in order to understand what a category is, you first have to know what a class is." (John L Bell, "Category theory and the foundations of mathematics", The British Journal for the Philosophy of Science 32(4), 1981)
"[…] it would be technically possible to give a purely category-theoretic account of all mathematical notions expressible within axiomatic set theory, and so formally possible for category theory to serve as a foundation for mathematics insofar as axiomatic set theory does." (John L Bell, "Category theory and the foundations of mathematics", The British Journal for the Philosophy of Science 32(4), 1981)
"It is a remarkable empirical fact that mathematics can be based on set theory. More precisely, all mathematical objects can be coded as sets (in the cumulative hierarchy built by transfinitely iterating the power set operation, starting with the empty set). And all their crucial properties can be proved from the axioms of set theory. [...] At first sight, category theory seems to be an exception to this general phenomenon. It deals with objects, like the categories of sets, of groups etc. that are as big as the whole universe of sets and that therefore do not admit any evident coding as sets. Furthermore, category theory involves constructions, like the functor category, that lead from these large categories to even larger ones. Thus, category theory is not just another field whose set-theoretic foundation can be left as an exercise. An interaction between category theory and set theory arises because there is a real question: What is the appropriate set-theoretic foundation for category theory?" (Andreas Blass, "The interaction between category theory and set theory", 1983)
"We consider forcing over categories as a way of constructing objects by geometric approximation, including a construction of a generic model of a geometric theory as its special case." (Andrej Scedrov, "Forcing and Classifying Topoi", Memoirs of the American Mathematical Society Vol. 48, 1984)
"Symmetries of a geometric object are traditionally described by its automorphism group, which often is an object of the same geometric class (a topological space, an algebraic variety, etc.). Of course, such symmetries are only a particular type of morphisms, so that Klein’s Erlanger program is, in principle, subsumed by the general categorical approach." (Yuri I Manin, "Topics in Noncommutative Geometry", 1991)
"[…] categorical logic is, to a great degree, autonomous, even in matters syntactical. ([203], 54.)" (Michael Makkai, "Generalized sketches as a framework for completeness theorems", Journal of Pure and Applied Algebra, 115(1), 1997)
"What was clearly useful was the use of diagrams to prove certain results either in algebraic topology, homological algebra or algebraic geometry. It is clear that doing category theory, or simply applying category theory, implies manipulating diagrams: constructing the relevant diagrams, chasing arrows by going via various paths in diagrams and showing they are equal, etc. This practice suggests that diagram manipulation, or more generally diagrams, constitutes the natural syntax of category theory and the category-theoretic way of thinking. Thus, if one could develop a formal language based on diagrams and diagrams manipulation, one would have a natural syntactical framework for category theory. However, moving from the informal language of categories which includes diagrams and diagrammatic manipulations to a formal language based on diagrams and diagrammatic manipulations is not entirely obvious." (Jean-Pierre Marquis, "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory", 2009)
"Category theory studies relationships between things and builds on this in various ways: characterising things by what properties they have, finding the pond in which things are the biggest fish, putting things in context, expressing subtle notions of things being ‘more or less the same’." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"In category theory there is always a tension between the idealism and the logistics. There are many structures that naturally want to have infinite dimensions, but that is too impractical, so we try and think about them in the context of just a finite number of dimensions and struggle with the consequences of making these logistics workable."
"This is how category theory arose, as a new piece of maths to study maths. In a way category theory is an ultimate abstraction. To study the world abstractly you use science; to study science abstractly you use maths; to study maths abstractly you use category theory. Each step is a further level of abstraction. But to study category theory abstractly you use category theory." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
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